Question: The day length in Manila, Philippines, varies over time in a periodic way that can be modeled by a trigonometric function. Assume the length of the year (which is the period of change) is exactly $365$ days long. The shortest day of the year is December $21$, and it's $675.85$ minutes long. Manila's longest day is $779.60$ minutes long. Note that December $21$ is $11$ days before January $1$. Find the formula of the trigonometric function that models the length $L$ of the day $t$ days after January $1$. Define the function using radians. $ L(t) = $ What is the day length on the People Power Anniversary (February $25$, which is $55$ days after January $1$ ) in Manila? Round your answer, if necessary, to two decimal places. $ $
Let's start by letting $u$ be the number of days since December $21$. Both sine and cosine can be used to model periodic contexts. We can decide which is better fitting by considering the $y$ -intercept. The sine function intercepts the $y$ -axis at its midline, and the cosine function intercepts the $y$ -axis at its peak. We know Manila's shortest day was December $21,$ when $u = 0$. Since $\cos u$ has a peak at $u = 0$, let's use a cosine function to model the length of the days in Manila. We'll have to flip it vertically to get a minimum at $u= 0$. The amplitude of the length of the days is $\dfrac{779.6-675.85}{2} = 51.875$ minutes. The period is $365$ days. The midline is the average of the highest and lowest values, or $\dfrac{779.6 + 675.85}{2} = 727.725$ Since the ordinary cosine function $f(u) = \cos u$ has period $2\pi$, midline $y = 0$, and amplitude $1$, we need to stretch it horizontally by a factor of ${\dfrac{365}{2\pi}}$, stretch it vertically by a factor of ${51.875}$, flip it vertically, and move it up ${727.725}$ units: $ L(u) = {-51.875}\cos\left({\dfrac{2\pi}{365}}u\right) + {727.725}$ Since December $21$ is $11$ days before January $1$, the day that is $t$ days after January $1$ is $t + 11$ days after December $21$, so $u = t +11$ : $ L(t) = {-51.875}\cos\left({\dfrac{2\pi}{365}}(t + 11)\right) + {727.725}$ Since February $25$ is $55$ days after January $1$, $ \begin{aligned}L(55) &= {-51.875}\cos\left({\dfrac{2\pi}{365}}(55 + 11)\right) + {727.725}\\ &\approx 705.88 \end{aligned}$ A correct formula for $H(t)$ is: $ L(t) = -51.875\cos\left(\dfrac{2\pi}{365}(t + 11)\right) + 727.725$ The day length on The People Power Anniversary is: $ 705.88$ minutes